When your high-school teacher asked you to ‘prove’ something in math-class, he was asking you to show that it all held together nicely. In other words, that you were not violating the Principal Principle, you were not contradicting yourself somewhere in the fine-print.
The Contradiction Principle demands that all derivations meet its condition of internal-consistency. But what then is the defending criterion for this celebrated Principle itself?
Why should you not violate the Principle of Contradiction?
Well, it goes like this. Here is Aristotle in Metaphysics:
‘The possibility of a middle between contraries is excluded: for it is necessary to assert or deny one thing or another. This is clear from the definition of Truth and Falsity: either what is, is affirmed or denied, or else what is not, is affirmed or denied, there can be no middle ground..
Similarly, every thought and concept is expressed as an affirmation or a negation, this is clear from the definition of Truth and Falsity.
Hence also, the frequent saying befalls all such arguments, that they destroy themselves. For he who says that all things are true presents even the statement contrary to his own as true, and therefore his own as not true: whereas he who says that all things are false presents also himself as false.’
Aristotle called such an appeal a ‘Self-Destroying Argument’. A perfect and precise phrase, vintage Aristotle.
Does it ring a loud bell?